Article Volume 11, Number 10 8 October 2010 Q10005, doi:10.1029/2010GC003227 ISSN: 1525‐2027
A noninvasive method for measuring the velocity of diffuse hydrothermal flow by tracking moving refractive index anomalies Eric Mittelstaedt and Anne Davaille Fluides, Automatiques et Systèmes Thermiques Laboratoire, CNRS/UPMC/UPS, F‐91405 Orsay CEDEX, France (
[email protected]‐psud.fr)
Peter E. van Keken Department of Geological Sciences, University of Michigan, Ann Arbor, Michigan 48109, USA
Nuno Gracias EIA Department, University of Girona, Ed. PIV, E‐17003 Girona, Spain
Javier Escartin Institut de Physique du Globe de Paris, Université de Paris 6, F‐75005 Paris, France [1] Diffuse flow velocimetry (DFV) is introduced as a new, noninvasive, optical technique for measuring the velocity of diffuse hydrothermal flow. The technique uses images of a motionless, random medium (e.g., rocks) obtained through the lens of a moving refraction index anomaly (e.g., a hot upwelling). The method works in two stages. First, the changes in apparent background deformation are calculated using particle image velocimetry (PIV). The deformation vectors are determined by a cross correlation of pixel intensities across consecutive images. Second, the 2‐D velocity field is calculated by cross correlating the deformation vectors between consecutive PIV calculations. The accuracy of the method is tested with laboratory and numerical experiments of a laminar, axisymmetric plume in fluids with both constant and temperature‐ dependent viscosity. Results show that average RMS errors are ∼5%–7% and are most accurate in regions of pervasive apparent background deformation which is commonly encountered in regions of diffuse hydrothermal flow. The method is applied to a 25 s video sequence of diffuse flow from a small fracture captured during the Bathyluck’09 cruise to the Lucky Strike hydrothermal field (September 2009). The velocities of the ∼10°C–15°C effluent reach ∼5.5 cm/s, in strong agreement with previous measurements of diffuse flow. DFV is found to be most accurate for approximately 2‐D flows where background objects have a small spatial scale, such as sand or gravel. Components: 10,100 words, 11 figures. Keywords: hydrothermal vents; diffuse flow; cross‐correlation techniques; particle image velocimetry. Index Terms: 3017 Marine Geology and Geophysics: Hydrothermal systems (0450, 1034, 3616, 4832, 8135, 8424); 3094 Marine Geology and Geophysics: Instruments and techniques; 0540 Computational Geophysics: Image processing. Received 17 May 2010; Revised 26 August 2010; Accepted 31 August 2010; Published 8 October 2010.
Copyright 2010 by the American Geophysical Union
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Mittelstaedt, E., A. Davaille, P. E. van Keken, N. Gracias, and J. Escartin (2010), A noninvasive method for measuring the velocity of diffuse hydrothermal flow by tracking moving refractive index anomalies, Geochem. Geophys. Geosyst., 11, Q10005, doi:10.1029/2010GC003227.
1. Introduction
velocity and temperature measurements to an entire vent field [e.g., James and Elderfield, 1996].
[2] Since the discovery of low‐temperature hydrothermal venting along the Galapagos Spreading Center in the late 1970s [Corliss et al., 1978; Edmond et al., 1979], several methods have been introduced to estimate the heat flux associated with hydrothermal activity and the partition between diffuse and discrete venting. In general, these methods involve either direct point measurements at the site of diffuse venting using invasive flow collectors [e.g., Ramondenc et al., 2006; Sarrazin et al., 2009; Schultz et al., 1992], or indirect methods that measure fluid properties (e.g., temperature) in the effluent plume after it buoyantly rises from the seafloor [e.g., Rona and Trivett, 1992] and/or is advected horizontally by ocean currents [e.g., Trivett and Williams, 1994; Veirs et al., 2006].
[4] In addition to the above sources of uncertainty, bathymetric variations and naturally occurring fauna can present difficulties for some direct measurement techniques. Direct measurement devices that use sensors in a vertical cylinder above a flow concentrator [Sarrazin et al., 2009; Schultz et al., 1992; Schultz et al., 1996] can accurately measure fluid velocities (e.g., ±8% [Schultz et al., 1992]). However, natural seafloor roughness and small fractures where diffuse flow is emitted [Baker et al., 1993; Rona and Trivett, 1992] can prevent sealing between the seafloor and the flow concentrator. As noted by Sarrazin et al. [2009], a similar problem is also presented by biological communities of tube worms that are often found on or near diffuse flow sources.
[3] Estimates of diffuse heat fluxes at individual
hydrothermal fields range from below the detection limit at the Rainbow vent field [German et al., 2010] to ∼2000 MW at the TAG hydrothermal field [Rona and Trivett, 1992]; a range that reflects interfield differences, temporal variability, scarcity of measurements, and uncertainty in measurement techniques. Published uncertainties in the diffuse heat flux, where stated, are commonly between ∼50% and ∼70% of the mean estimated heat flux [James and Elderfield, 1996; Ramondenc et al., 2006; Rona and Trivett, 1992; Trivett and Williams, 1994]. Measurement scarcity is a significant source of uncertainty regardless of the measurement technique and can only be improved by further field studies with currently available techniques. Other sources of uncertainty for indirect methods include unconstrained plume sizes [e.g., Trivett and Williams, 1994], mixing of different vent sources [e.g., Baker et al., 1993; Ginster et al., 1994], unknown temporal variability of the diffuse flux at the source [e.g., Scheirer et al., 2006; Sohn, 2007], and modeling assumptions [e.g., Rona and Trivett, 1992; Trivett, 1994]. For direct methods, sources of uncertainty often include the effect of invasive measurement devices on the flow [Ramondenc et al., 2006; Sarrazin et al., 2009; Schultz et al., 1992], unknown flow variability with time [e.g., Scheirer et al., 2006; Sohn, 2007], and extrapolation of small point‐like
[5] The above uncertainties and natural obstacles to current measurement methods suggest that a technique which is accurate, capable of surveying large areas of seafloor, can perform measurements on archived data, and is also flexible with regard to the terrain type will provide a useful addition to the already available methods. With this aim, we present a new optical technique, diffuse flow velocimetry (DFV), that measures the velocities of clear, diffuse, hydrothermal fluids by tracking of refractive index anomalies related to changes in fluid density (commonly due to differences in temperature and/ or salinity). Here, we introduce the DFV method and demonstrate its accuracy in laboratory and numerical tests. We then discuss the applicability of DFV to diffuse hydrothermal flow, compare DFV to previous measurement techniques, and present an example calculation from a fracture at the Tour Eiffel vent site in the Lucky Strike hydrothermal field on the Mid‐Atlantic Ridge.
2. Diffuse Flow Velocimetry 2.1. Laboratory Imaging of Index of Refraction Anomalies [6] Variations in fluid density produce variations in refractive index [Gladstone and Dale, 1863] so that the trajectory of a light ray traversing an anoma-
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lously dense fluid volume bends in the direction of increasing density. Objects viewed through the lens of such an index of refraction anomaly appear distorted (e.g., the “waves” seen above hot concrete on a summer day). There are a number of well‐ established measurement techniques that take advantage of this property. These include the schlieren laboratory technique [Töpler, 1866] and, recently, several correlation‐based imaging techniques including “synthetic schlieren” [Dalziel et al., 2000; Sutherland et al., 1999] and “background‐ oriented schlieren” [Meier, 2002]. Synthetic and background‐oriented schlieren are used both inside and outside of the lab to quantitatively measure variations in fluid density associated with, for example, moving helicopter blades [Richard and Raffel, 2001], the turbulent mixing of helium gas [Meier, 2002], sound waves from a gunshot [Meier, 2002], and an oscillating bar in a linearly stratified fluid [Sutherland et al., 1999]. [7] DFV builds upon the above correlation‐based
methods which have their roots in the classical schlieren technique. Briefly, in the classic schlieren technique a knife edge is placed near the focal point of a parabolic mirror as an asymmetric aperture to block a fraction of previously collimated light after it traverses an index of refraction anomaly. Light intensity variations in the resulting image, due to bent light rays, are related to gradients in fluid density. Synthetic and background‐oriented schlieren techniques measure variations in fluid densities without the need for the complicated mirrors and lighting of standard schlieren. The apparent shift of light rays passing through an index of refraction anomaly is calculated by cross correlating the image of a reference background (placed behind a uniform fluid at rest) with the resulting image when the background is seen through a density anomaly. The result is a 2‐D field of apparent background deformation associated with density gradients in the fluid. In the case of a moving fluid, the apparent deformation field is displaced along with the density gradients. The new method presented here, diffuse fluid velocimetry (DFV), exploits the connection between the displacement of apparent background deformation and fluid movement to determine the 2‐D velocity field.
2.2. Diffuse Flow Velocimetry Method [8] We discuss DFV in the application of hydrothermal flow at the seafloor. Video sequences
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provide a series of images of a motionless, random medium (e.g., rocks) obtained through the lens of a moving refraction index anomaly (e.g., a hot upwelling) (Figure 1a). A DFV calculation involves two steps (Figure 2); the first determines the change in apparent deformation of subsequent background images and the second determines the movement of the apparent deformation through time to determine the fluid velocities. In the first step of the DFV method, the deformation field is determined using particle image velocimetry (PIV), for which we use the LaVision DaVis software (http://www.lavision. de/en/), as discussed by Davaille and Limare [2007] and Limare et al. [2008]. In the PIV calculation, a pair of consecutive images at times t0 and t1 are each divided into overlapping windows of L by N pixels. Here we use 8 × 8 to 32 × 32 pixels windows with an overlap of 50%. Using Fourier convolution, intensities at vertical pixel location l and horizontal pixel location n from a window at time t0, Ilnt0 are cross correlated with the spatially t1 shifted window at time t1, Il+i,n+j , to create a correlation matrix Cij Cij ¼
L X N X
t1 Ilnt0 *Ilþi;nþj ;
ð1Þ
l¼1 n¼1
where the indices i and j correspond to vertical and horizontal pixel shifts of the window at t1. The location of the maximum value in the correlation matrix corresponds to the highest probability displacement of the window caused by the refraction index anomaly between t0 and t1. This procedure is repeated for all windows and for all subsequent image pairs (i.e., t0 and t1, t1 and t2, …, tn–1 and tn). The result is an instantaneous 2‐D vector field of the change in the apparent background deformation due to movement of the index of refraction anomaly between each image pair (Figures 2d and 2e). For brevity, these changes in apparent background deformation are hereafter referred to as deformation vectors or, more generally, deformation. [9] The deformation computed by DFV is different from that obtained by schlieren techniques where the correlation is performed in reference to a fixed, undistorted image. In DFV, the deformation vectors are computed from changes in the apparent background deformation between two subsequent images. This is essential in a hydrothermal environment where a reference image of the seafloor is unavailable due to continuous fluid flow. An estimate of the static background image can be
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Figure 1. Diffuse flow velocimetry (DFV) calculates velocities from (a) a series of images of motionless, random media (e.g., rocks or sand) obtained through moving refraction index anomalies. Tests of DFV on an upwelling, laminar, axisymmetric plume are performed with (b) numerically distorted images created by shooting rays through numerical temperature fields and (c) images taken of a random dot pattern in the background of a laboratory tank. (d) Refraction index anomalies in both tests are caused by the linear relationship between index of refraction and temperature for silicone oil (numerical) and glucose syrup (laboratory).
obtained from the temporal median of pixel intensities, but is very prone to blurriness and loss of detail which degrade correlation accuracy.
of both the horizontal and vertical components, X and Y, of the deformation vectors
[10] In a moving fluid, the deformation field moves
a distance d between time t0.5 and t1.5 (the average times between t0 and t1 and t1 and t2 between which the deformation field is calculated as above). If we assume that in the small time between t0.5 and t1.5 the shape of the density gradients in the fluid remains nearly unchanged, d can be determined by cross correlation of the deformation vectors. In the second step of the DFV method, two subsequent deformation vector fields at t0.5 and t1.5 are each divided into overlapping windows of L by N deformation vectors. We use windows of 8 × 8 vectors with a 50% overlap. For a single window, the correlation matrix Dij is defined to be a function
Dij ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP 2 N uL P t1:5 u X t0:5 Xlþi;nþj tl¼1 n¼1 ln
þ
L*N vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP 2 L N u P t0:5 t1:5 u Yln Ylþi;nþj tl¼1 n¼1 L*N
;
ð2Þ
where l and n are the local vertical and horizontal window coordinates of a deformation vector, and i and j are the vertical and horizontal window vector shifts. The location of the minimum of Dij defines the highest probability displacement vector d of the window between t0.5 and t1.5. The precision of the location of the
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Figure 2. A cartoon representation of the steps involved in DFV. (a–c) Apparent deformation of a background in subsequent images is captured through the lens of a refractive index anomaly (ellipses). (d and e) The change in the apparent background deformation is calculated between these images (vectors). Finally, the location of the calculated deformations is tracked between calculations to yield (f) the fluid velocities (vectors).
correlation minimum is improved from ±0.5 times the distance between vector locations to ∼±0.1 times the intervector distance with an analytical three‐point Gaussian fit in both coordinate directions [Willert and Gharib, 1991]. The use of the Gaussian fit is motivated by its rapid calculation and its low interpolation errors in PIV applications [Roesgen, 2003]. The displacement vector d yields a velocity vector for the given window *
v¼
dy py S dx px S ^x þ ^y; ðt1:5 t0:5 Þ ðt1:5 t0:5 Þ
ð3Þ
where px and py are the horizontal and vertical number of pixels between each deformation vector, S is the ratio of centimeters to pixels, * and v is the velocity vector in cm/s at time t1. The cross correlation is performed on all the windows to yield the instantaneous, 2‐D velocity field (Figure 2f). [11] The location of the correlation minimum in
equation (2) gives the highest probability displacement of the deformation field in the window, but outliers can occur due to poor image quality,
little or no fluid movement, and/or undetectable deformation (due to very small, very large, or nonexistent density variations). Three methods are used to limit false correlations. First, the velocity is considered valid only if the curvature in the immediate neighborhood of the correlation minimum is greater than an empirically determined critical value of 1 × 10−3 to 1 × 10−4. Second, a correlation minimum is considered invalid if it falls on the boundary of the correlation matrix. If a correlation minimum does not have a sufficient curvature or falls on the edge of the correlation matrix, it is assumed to be erroneous and the velocity in that location is set to 0. Finally, the calculated velocities are smoothed by a 3 × 3 median filter. Although some accurate velocities are eliminated, the combination of these three methods effectively reduces the number of highly inaccurate velocities (Figure 3). The above methods for eliminating false correlations produce similar results to methods using a local roughness criteria [e.g., Crone et al., 2010]. [12] Two sets of Matlab code are provided to
facilitate the above DFV calculations (auxiliary 5 of 18
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velocities with DFV. The results are compared with known plume velocities (laboratory PIV velocities and numerical model velocities). For a full description of the experimental methods see Appendix A. [14] Values of the Rayleigh and Prandtl numbers in
our experimental tests (Ra ∼103–4; Pr ∼104) differ from those expected in diffuse hydrothermal systems (Ra ∼105–6; Pr ∼101). However, these numbers describe dynamical conditions of the flow, not the magnitude of apparent background deformation critical to DFV measurements. A more relevant value is the theoretical scale of apparent background deformation (Appendix B)
Figure 3. The straightforward correlation of deformation vectors produces velocities that can have large errors (light gray bars), but the use of filtering and ensuring a minimum curvature in the correlation minimum effectively eliminates the most erroneous vectors (dark gray bars).
material).1 The first calculates the apparent changes in background deformation between images and is a slightly modified version of OpenPIV (http:// www.openpiv.net/). Although we use the proprietary DaVis software to perform PIV calculations here, final DFV velocity fields are comparable when using OpenPIV. The second is the DFV code used by the authors to calculate fluid velocities from deformation vectors.
3. Experimental Tests
Dp 2dB DT @n ¼ ; h hnambient @T
where h is the characteristic half‐width of the index of refraction anomaly, Dp is the vertical shift of a light ray from horizontal, and dB is the distance between the anomaly and the image capture device (Figure 5). Values of Dp/h range from 0.1 to 2.1 in our experiments and are estimated to be ∼0.5–0.6 for the example of hydrothermal diffuse flow in section 5.4, within the range of our experimental conditions (Figure B1).
4. Results 4.1. Observed Patterns of Apparent Background Deformation [15] The nearly identical setups of the numerical
[13] The accuracy of DFV velocities is tested using
two experiments. The first is based in the laboratory and the second uses a numerical model based upon a similar laboratory setup. In both experiments, a thermally buoyant, axisymmetric, laminar plume produces the required refractive index anomaly (Figures 1b and 1c). To introduce the upwelling plume, a fluid (sugar syrup in the laboratory and silicone oil for the numerical method) is heated from below by a localized, circular heat source. The temperature gradient between the plume and the ambient fluid produces refraction index gradients that distort images of a random (laboratory) or regular (numerical) dot pattern behind the plume. In the laboratory, images are captured using a red‐filtered light source and camera. We use ray tracing to construct synthetic images from the numerical results. Images of the distorted backgrounds are used to calculate fluid 1 Auxiliary materials are available in the HTML. doi:10.1029/ 2010GC003227.
and laboratory experiments produce similar patterns of deformation (Figures 4b and 4e). Note, however, that the sign of the deformation in the laboratory images is reversed with respect to the numerical results discussed below due to the different optical focus plane locations (Figure 5). This difference does not affect the DFV calculation. Initially, formation of a conductive thermal boundary layer at the base of the tank or numerical domain produces very strong temperature gradients that distort the background image to the point of being unrecognizable (i.e., deformations cannot be successfully calculated in this region). After convective motions begin and the upwelling leaves the boundary layer, the temperature gradients decrease and are largest on the edges of the rising plume head and tail (i.e., between the warm, upwelling plume and the cooler, ambient fluid). Around the plume head, deformations form two horseshoe‐ shaped regions, one just above the other with the lower region appearing to deform in the negative 6 of 18
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Figure 4. An upwelling thermal plume acts as a refraction index anomaly causing an apparent distortion of (a) a numerically produced grid (t = 100 s) and (d) the background of a laboratory tank (t = 110 s). The calculated deformation magnitudes of the (b) numerical and (e) laboratory background are on the order of 0.01 cm. The DFV velocities (red arrows) are calculated by cross correlating the deformation field between successive calculations and are compared with (c) numerical velocities (black arrows) or (f) PIV velocities (black arrows) of particles imaged in the flow.
direction and the upper region in the positive direction (Figure 4b). The inversion in the sign of deformation is caused by the divergent refraction anomaly initially “moving” the background image upward and outward (positive deformation) and the subsequent return of the image to a less distorted state, moving downward and inward (negative deformation), as the anomaly passes (Figures 2d, 2e, 6a, and 6b). The magnitude of the background deformations in the plume head region is on the order of 0.01 cm, but is zero in the plume stem after it achieves a steady state temperature structure. As the plume head continues to rise, the magnitude of the deformations decreases with the decreasing temperature anomaly (Figures 6a and 6b). This suggests that the anomaly will become undetectable, although at a scale height larger than that seen in our experiments. Overall, we see less uniformity in the numerical deformation pattern than that of the laboratory experiment, in part due to discretization errors in the generation of the numerical images (auxiliary material).
4.2. Advection, Diffusion, and Fluid Velocities [16] The DFV velocities of the thermal plume in
our experiments are not equal to particle velocities
within the fluid, but are due to a combination of fluid advection and thermal diffusion. Centerline profiles (i.e., along the vertical plume axis) of the vertical temperature gradient (dT/dz), the vertical deformation, and contours of temperature from the numerical experiment at four times (t1, t2, t3, and t4) show that the height of the maximum and
Figure 5. The original positions of pixels (green dots) seen through a divergent refraction index anomaly appear to move toward each other (black circles) due to the location of the optical focus plane. The numerical experiment calculates the position of the light rays at the location of the image capture device and, thus, displays the opposite pixel motion. See Appendix B for a description of parameters. 7 of 18
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Figure 6. Centerline profiles of (a) the vertical temperature gradient at four times t1 (black dashed line), t2 (gray dashed line), t3 (solid black line), and t4 (solid gray line); (b) calculated deformation between t1 and t2 (dashed line, Dt = 5 s) and t3 and t4 (solid line, Dt = 5 s); and (c) isotherms at each time show that the locations of deformation extremes lie just below the locations of the maximum temperature gradients (gray bars). (d) The relative location of the deformation extremes (gray dashed lines) and the maximum vertical temperature gradients (black lines) is nearly constant. The deformation extremes also lie below the location of the plume head stagnation point where the radial velocity gradient is a maximum. (e) For most of the experiment, the velocity of the stagnation point (black line) and the maximum vertical temperature gradient (gray line) are almost identical.
minimum deformations between t1 and t2 and between t3 and t4 (Dt = 5 s) fall just below the heights of the maximum temperature gradients (dT/dz) at each time (Figures 6a–6c). The relative positions of the maximum dT/dz, controlled by advection and thermal diffusion, and the extremes in deformation are constant throughout nearly the entire risetime of the upwelling plume (Figure 6d), indicating identical velocities. The contribution of thermal diffusion is reinforced by a common laboratory measure of plume head velocity; the move-
ment of the stagnation point at the top of the plume head where the radial velocity gradient dVr/dr is a maximum. For similar experimental fluids, the velocity of the stagnation point is previously shown to include advection as well as conductive expansion of the plume head [Davaille et al., 2010]. The location of the stagnation point in our numerical model is slightly above the location of the maximum temperature gradient and between the extremes in deformation throughout the risetime of the plume (Figures 6d and 6e). Thus, the velocity of the 8 of 18
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Figure 7. (a) The distance between two initially parallel rays (black dots, ray separation, negative if the rays cross) shot through the numerical thermal field is linearly related (gray line) to the distance (measured from the plume center) to the image plane. If the rays exit the plume at slightly different angles, they may cross (crossing point, ray separation = 0). (b) The distance from the plume center to the ray crossing (black dots) is a strongly nonlinear function of the vertical temperature gradient at the ray location as it crosses the plume center.
extremes in deformation (i.e., the DFV velocity) is identical to the velocity of the stagnation point.
4.3. Image Distortion and Ray Crossings [17] Two closely spaced, parallel rays traversing a
fluid volume with large gradients in refractive index may bend to slightly different degrees and eventually cross after they have left the anomalous fluid. Crossing light rays may partially obscure background images and add noise to deformation calculations. To examine this possibility, we shoot two initially parallel rays (Appendix A), separated vertically by 0.025 cm, through the center of the plume axis and examine the final ray separations at different distances beyond the plume center (Figure 7a). After leaving the thermal anomaly, the
index of refraction remains constant and the ray separation is a linear function of distance from the plume center. For cases where the rays cross, the vertical temperature gradient in the plume center (at the height of the ray) is used as a proxy for the intensity of the index of refraction gradients along the ray trajectory (Figure 7b). Ray crossing distance is a strongly nonlinear function of the vertical temperature gradient with distances between 3.3 cm and 40.3 cm for vertical thermal gradients of between 198.6°C/cm and 4.51°C/cm. The crossover distances for similar thermal gradients in seawater are likely to be larger because the temperature dependence of the index of refraction is approximately one third that of silicone oil (dn/dT = ∼−0.9 × 10−4 K−1 best fit for seawater between 5°C and 30°C at a wavelength of 589 nm [Quan and 9 of 18
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Figure 8. Misfit between DFV calculated velocity and the (a) stagnation point velocity scaled by the stagnation point velocity and (b) the RMS difference between the DFV velocities and material velocities scaled by the maximum velocity. The DFV velocities are compared to both numerical (black crosses) and laboratory PIV velocities (red circles).
Fry, 1995] and dn/dT = −3.71 × 10−4 K−1 for silicone oil).
4.4. Misfit in Experimental DFV Velocities [18] The DFV velocity vectors represent an average
velocity of the thermally induced refraction index anomaly within each window. For the conditions of our experiments, the difference between this velocity and the velocity of particles within the fluid is ∼5%–10% [Davaille et al., 2010] (section 4.2). Thus, the particle velocities are a poor metric for quantitative errors in the DFV calculations. Instead, quantitative errors are determined using the stagnation point velocity, which moves with the refractive index anomaly (section 4.2). The stagnation point velocity, however, only provides error measurements at a single location and the differences between the numerical and laboratory PIV and the DFV velocities are still used for qualitative discussion of the error. [19] Quantitative errors are calculated by finding the
misfit between the stagnation point velocity and the DFV velocity interpolated to the same location and scaled by the stagnation point velocity (Figure 8a). The misfit for the numerical cases varies from 0.005 to 0.418 and for the laboratory cases from 0.008 to 0.382. The RMS error of all the numerical measurements is 0.1705 and for all the laboratory measurements is 0.1656. [20] During the initial stages of plume upwelling
the errors tend to be high, but later settle to a lower level (Figure 8a). This may be due to the initially high temperature gradients (i.e., large background
deformations) in the plume head as it leaves the thermal boundary layer (auxiliary material). In the numerical experiment, the RMS error after and including 80 s is 0.067, a decrease of 0.104 from the overall value. In the laboratory, the RMS error after and including 90 s is 0.05, a decrease of 0.116. These lower values are more representative of the level of error seen in the established flow during the majority of the experiments (70% of the numerical experiment and 56% of the laboratory experiment). [21] More qualitatively, the DFV velocities can be
compared with local spatial averages of the numerical or PIV values within the area of each window. In general, the DFV and the locally averaged numeric and PIV vectors closely resemble one another with many vectors nearly completely overlapping (Figures 4c and 4f). For a given time, we calculate the root‐mean‐square (RMS) difference between the total velocities V of the DFV and the numerical or laboratory calculations scaled by the maximum velocity (Figure 8b). For the numerical experiment, the RMS differences are between 0.1511 and 0.359 and for the laboratory experiment are between 0.087 and 0.426. As for the comparison with the stagnation point velocity, an initially high difference is followed by a decrease to a lower level. In the numerical experiment, the temporally averaged RMS error is 0.213, but taking only the values after and including 80 s, it falls to 0.174 (decrease of 0.04). Similarly, the average RMS difference over all time from the laboratory experiment is 0.208, and after 78 s it becomes 0.188 (decrease of 0.02). 10 of 18
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Figure 9. Contour plots of the difference between the magnitude of DFV velocities and numeric or PIV velocities as a percentage of the maximum velocity for (a) the numerical and (b) the laboratory tests. A region of very small differences (
